Question

In the figure, the inner (shaded) region $A$ represents a sphere of radius $r_A=1$, within which the electrostatic charge density varies with the radial distance $r$ from the center as ${\rho }_A=kr$, where $k$ is positive. In the spherical shell $B$ of outer radius $r_B$, the electrostatic charge density varies as ${\rho }_B=\frac{2k}{r}$. Assume that dimensions are taken care of. All physical quantities are in their SI units.

Which of the following statement(s) is(are) correct?

• A. If $r_B=\sqrt{\frac{3}{2}}$, then the electric field is zero everywhere outside $B$.
• B. If $r_B=\frac{3}{2}$, then the electric potential just outside $B$ is $\frac{k}{{ϵ}_0}$.
• C. If $r_B=2$, then the total charge of the configuration is $15\pi k$.
• D. If $r_B=\frac{5}{2}$, then the magnitude of the electric field just outside $B$ is $\frac{13\pi k}{{ϵ}_0}$.

Answer 1

Answer: (B)

$q_1=\underset{0}{\overset{1}{\int }}kr4\pi r^{2}dr=\frac{4\pi k}{4}=\pi k$
$q_2=\underset{1}{\overset{r}{\int }}\frac{2k}{r}4\pi r^{2}dr=\frac{8\pi k\left(r^2-1^2\right)}{2}$
$q_2=4\pi k\left[r^2-1\right]=4\pi kr{r}^2-4\pi k$
$q_{net}=q_1+q_2$
$=4\pi k{r}^2-3\pi k$
$q_{\text{net }}=\pi k\left[4r^2-3\right]$
(A) $E_{\text{net }}=0\Rightarrow q_{\text{net }}=0\Rightarrow r=\frac{\sqrt{3}}{2}$
(B) $V=\frac{k{Q}_{net}}{r}=\frac{1}{4\pi {\epsilon }_0}\frac{\pi k\left(4r^2-3\right)}{r}$
$V=\frac{k}{4{\epsilon }_0}\left[4r-\frac{3}{r}\right]$
$=\frac{k}{4{\epsilon }_0}\left[4\times \frac{3}{2}-\frac{3\times 2}{3}\right]=\frac{k}{{\epsilon }_0}$
(C) $q_{net}=\pi k\left[4(2{)}^2-3\right]$
$=13\pi k$
(D) $E_2=\frac{kQ}{r^2}$
$=\frac{1}{4\pi {\epsilon }_0}\frac{\pi k\left(4r^2-3\right)}{r^2}$
$=\frac{k}{4{\epsilon }_0}\left[\frac{4{\left(\frac{5}{2}\right)}^2-3}{(5/2{)}^2}\right]$
$=\frac{k}{25{\epsilon }_0}[25-3]=\frac{22}{25}\frac{k}{{\epsilon }_0}$